Вычислить значение производной сложной функции u=u(x,y), где x=x(t), y=y(t), при t=t0 с...

5.9.
$u=x^2 * e^{-y} \\ x=sint \\ y = sin^2 t \\ \\ u = sin^2 t * e^{-sin^2 t} \\ \\ \frac{du}{dt} = (sin^2 t)' * e^{-sin^2 t} +sin^2 t * (e^{-sin^2 t} )' = \\ \\ = 2*sint*cost*e^{-sin^2 t}+ sin^2 t *e^{-sin^2 t}*(-2*sint)*cost = \\ \\ = 2*sint*cost*e^{-sin^2 t}*(1-sin^2 t) = \\ \\ = 2*sint*cos^3t*e^{-sin^2 t} \\ \\ sint_0 = sin \frac{ \pi }{2} = 1 \\ cost_0 = cos \frac{ \pi }{2} \\ \\ \frac{du}{dt} (t_0) = 2*1*0^3 *e^{-1} = 0$